Two hawks show up in the writings for Brokeback Mountain:
One is mentioned by Annie Proulx as she says that in 1963:
"There were only the two of them on the mountain flying in the euphoric, bitter air, looking down on the hawk's back and the crawling lights of vehicles on the plain below, suspended above ordinary affairs and distant from tame ranch dogs barking in the dark hours."
In the screenplay a hawk is seen in 1982:
"We pull back and see the great sweep of the bald, desolate countryside. Ennis passes several abandoned ranch houses sitting blank-eyed, surrounded with weeds. The corral fences have fallen down. A lone hawk circles. Ennis slows down."